Lab 2 - High-Frequency Models

Microstructures and Trading Systems



Ricardo Figueroa Acosta, facosta@iteso.mx

06.2022 | Repository: Link

High-Frequency Trading Models

Asset Pricing Theory (APT-Martingale) - Roll Model

Abstract

High-Frequency trading models are an interesting proposal in quantic finance, since its a different approach from what traditional trading and finance means; in this project we are going to dive deep into two of the most know High-Frequency trading models, from the Asset Pricing Theory: Martingale; which states that, from a market microstructure perspective, a price in the future will be exactly the same as the present price; i.e: $p_{t+1} = p_t.$

Keeping the same market microstructure perspective, we have the Roll model (1984) which... </font>

1. Introduction

In this project we are going to dive deep into two of the most known High-Frequency Trading Models; the Asset Pricing Theory which statets that the price of a risky asset, $P_t$, from a consuming model in present and future time, under the perspective of the market microstructure, can be modeled as a stochastic process named martingale.

A martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time (in seconds or milliseconds), in this case, from the perspective of the market microstructure, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values i.e.: $p_{t+1} = p_t$.

This is going to be probed using an orderbook containing about 2400 timestamps. We are going to divide and group by minutes and count occurrences for martingales and non martingales.

The second model is the Roll Model (1984)...

2. Install/Load Packages and Depedencies

2.1 Python Packages

In order to run this notebook, it is necessary to have installed and/or have the requirements.txt file with the following:

  • numpy>=1.2
  • pandas>=1.4
  • plotly>=5.8

2.2 Files Dependencies

  • data.py: ordered data.
  • functions.py: functions used to solve main problems and perform experiments.
  • main.py: main script to call functions, visualizations and gather results.
  • visualizations.py: functions used to plot results.
  • requirements.txt: all libraries used and necessary to run this project.
  • orderbooks_05jul21.json: Orderbook containing raw data used in the project.

2.3 Libraries and scripts

In [ ]:
import plotly.io as pio
pio.renderers.default='notebook'

import main as mn
import functions as fn
import visualizations as vz

3. Data Description

orderbooks_05jul21.json:

Is an dataset that contains about 2400 orderbooks from bitfinex broker, split by dictionaries, which its primary key is a different timestamp. Every timestamp contains the following:

  • 'ask_size'
  • 'ask'
  • 'bid'
  • 'bid_size'

4. First experiment - APT

In [ ]:
help(fn.experiments)

Help on function experiments in module functions:

experiments(ob_data: dict, ob_ts: list, method: str) -> pandas.core.frame.DataFrame
    Function used to perform experiments with orderbook data.
    
    arguments:
    ----------
    ob_data: dictionary
    dictionary type with the following structure:
    'timestamp'
    'bid_size'
    'bid'
    'ask'
    'ask_size'
    
    ob_ts: list
    list with timestamps in string format.
    
    method: str: 'midprice' or 'wmidprice'
    string with the method that's going to be used in calculations.
    
    Returns -> dataframe 
    
    References:
    ----------
    [1] Martingale. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Martingale&oldid=49256

4.1 Experiment with midprice

In [ ]:
e1 = mn.e1
In [ ]:
e1[:3]
Out[ ]:
intervalo total e1 e1_proportion e2 e2_proportion
0 0 41 27 0.6585 14 0.3415
1 1 40 32 0.8000 8 0.2000
2 2 39 27 0.6923 12 0.3077
In [ ]:
e1.tail(3)
Out[ ]:
intervalo total e1 e1_proportion e2 e2_proportion
57 57 41 29 0.7073 12 0.2927
58 58 40 27 0.6750 13 0.3250
59 59 39 31 0.7949 8 0.2051

4.2 Demonstration

In order to probe if the midprices or weighted midprices follow a stochastic process, a martingale, it is necessary to probe the following statement: $p_{t+1} = p_t$.

Where

$p_{t+1} = \text{future price}$

and

$p_{t} = \text{present price}$

It is easy to probe by using a list comprehension

True_False_list = [midprice[i+1] == midprice[i] for i in range(len(midprice)-1)]

In [ ]:
print('The average proportion of martingale is', mn.e11_mean, 'and for no-martingales:', mn.e12_mean)

The average proportion of martingale is 0.73 and for no-martingales: 0.27

4.2 Experiment with weighted midprice

In [ ]:
e2 = mn.e2
In [ ]:
e2[:3]
Out[ ]:
intervalo total e1 e1_proportion e2 e2_proportion
0 0 41 27 0.6585 14 0.3415
1 1 40 27 0.6750 13 0.3250
2 2 39 26 0.6667 13 0.3333
In [ ]:
e2.tail(3)
Out[ ]:
intervalo total e1 e1_proportion e2 e2_proportion
57 57 41 27 0.6585 14 0.3415
58 58 40 27 0.6750 13 0.3250
59 59 39 26 0.6667 13 0.3333
In [ ]:
print('The average proportion of martingale is', mn.e21_mean, 'and for no-martingales:', mn.e22_mean)

The average proportion of martingale is 0.68 and for no-martingales: 0.32

The proportions change within the midprice and weighted midprice, but the martingale proportions is sill the highest one

5. Second experiment - Roll Model

In [ ]:
# DELETE THIS CELL IN YOUR FILE

# Another particular process that is relevant for the work. Normally, you include between 2 and 5 at the most.

6. Results

6.1 Results of first experiment

In [ ]:
help(vz.exp1_plot)

Help on function exp1_plot in module visualizations:

exp1_plot(df: pandas.core.frame.DataFrame, x: str, y: str) -> 'stackedbarplot'
    Function used to plot a stacked bar for experiment 1
    
    arguments:
    ----------
    df: DataFrame
    DataFrame containing results from a martingale analysis
    
    x: str
    x-axis (0-59 minutes)
    
    y: str
    y-axis (martingales and non martingales)
    
    Returns -> stacked barplot

6.1.1 Experiment with midprice

In [ ]:
e1_plot = vz.exp1_plot(df = e1, x = 'intervalo', y = ['e1', 'e2'])

The result is consistent in the whole hour, but less consistent than the weighted midprice result

6.1.2 Experiment with weighted midprice

In [ ]:
e2_plot = vz.exp1_plot(df = e2, x = 'intervalo', y = ['e1', 'e2'])

The result is consistent in the whole hour.

By observing the two results, we can conclude than in a microstructure market perspective, about 70% of the data will follow an stochastic process: a martingale.

6.2 Results of second experiment

In [ ]:
# DELETE THIS CELL IN YOUR FILE

# You can have several processes with which you are obtaining the results you need.

7. Conclusion

APT

The APT model is kind of abstract, it has many concepts and may seem kind of advanced and difficult to understand, but when diving a little more deep into its foundations and focusing on them, it is easy to understang its porpuse or what it explicit means: The best that we can expect in the future its what already is happening in the present; I thin it has much more sense from the market microstructure perspective, i.e. when time intervals are very small. As probed in this project, about 70% a price, in the market microstructure perspective, will be a martingale.

Roll

x

8. References

[1] Munnoz, 2020. Python project template. https://github.com/iffranciscome/python-project. (2021).

[2] Martingale. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Martingale&oldid=49256